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D₃×AGL(1,𝔽₅)×#104 #72229747

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

D₃×AGL(1,𝔽₅)×#104

Splittings

Fitting	C₃×C₅×#104 ⋊ C₂×C₄
Hall	C₃×C₅ ⋊ C₂×C₄×#104
Sylow	C₅ ⋊ C₄×D₃×#104
Sylow	C₃ ⋊ C₂×AGL(1,𝔽₅)×#104

Subgroups

	subgroup	G/H
center	C₄	D₃×D₄×AGL(1,𝔽₅)
commutator	C₃×C₄×C₅	C₂²×C₄²
Frattini	C₂×C₄	C₂²×D₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#104	C₂×C₄
socle	C₂×C₃×C₅	C₂×C₄×C₂²⋊C₄
prime radical	D₃×D₅×#41	C₂²
cosocle	C₂×C₃×C₄×D₅	C₂⁴
hypercenter	#104	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₂×C₄×#104
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

3756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 4 generators on 16 points:

(2,3,4,5)(6,7,8);(1,2,5,3,4)(7,8);(10,13,14,15);(9,10)(11,13)(12,14)(15,16)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₅)×#453
maximal normal subgroup of	D₃×AGL(1,𝔽₅)×#453

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3,4,5)(6,7,8), (1,2,5,3,4)(7,8), (10,13,14,15), (9,10)(11,13)(12,14)(15,16));